Cette histoire me donne des frissons.

Because histoire is a feminine singular noun.

French demonstratives change to match the noun:

• ce = this/that + masculine singular
• cet = this/that + masculine singular before a vowel or mute h
• cette = this/that + feminine singular
• ces = these/those + plural

So:

• ce livre = this book
• cet homme = this man
• cette histoire = this story

Even though histoire begins with h, it is still feminine, so cette is the correct form.

Here, histoire means story.

French histoire can mean:

• story
• history

The context tells you which one is meant. In Cette histoire me donne des frissons, it clearly means this story, not history.

Me means to me or me.

It is an indirect object pronoun placed before the verb.
So me donne literally means gives to me.

Breakdown:

• Cette histoire = this story
• me donne = gives me / gives to me
• des frissons = chills / shivers

So the structure is very close to English:
This story gives me chills.

In French, object pronouns usually go before the conjugated verb.

So French says:

• Il me parle. = He speaks to me.
• Tu me vois. = You see me.
• Cette histoire me donne des frissons. = This story gives me chills.

English often puts me after the verb, but French normally puts pronouns like me, te, lui, nous, vous, leur before it.

Donne is the third-person singular present tense of donner (to give).

The subject is Cette histoire, which is singular, so the verb must also be singular:

• je donne
• tu donnes
• il / elle / on donne

Since cette histoire is like it, French uses the il/elle form: donne.

Not really. It is a very natural expression, but it is somewhat idiomatic.

Literally, donner means to give, and the sentence literally looks like:

This story gives me shivers.

That works in English too, so the French is not strange. It means the story causes a physical reaction such as fear, tension, excitement, or unease.

So donner is still give, but in a very common figurative use.

Des frissons means shivers, chills, or goosebumps-like sensations.

A frisson is a small shiver that can come from:

• fear
• suspense
• emotion
• excitement
• cold

In this sentence, it usually suggests that the story is creepy, disturbing, or intense.

French normally uses frissons in the plural in this kind of expression, just as English often uses chills in the plural.

So:

• avoir des frissons = to have chills/shivers
• donner des frissons à quelqu’un = to give someone chills

The singular un frisson exists, but in expressions about a bodily reaction, the plural is very common and natural.

Because des is the normal indefinite plural article here.

It means something like some or just plural chills/shivers in a general sense.

• des frissons = chills / some shivers
• les frissons = the chills, referring to specific chills already known
• de frissons would not work here by itself

So donner des frissons is the standard expression.

Yes, but the meaning changes slightly.

• Cette histoire me donne des frissons = This story gives me chills.
• Cette histoire donne des frissons = This story gives chills / is chilling.

Without me, the sentence becomes more general. With me, you clearly say who is affected.

It can be both, depending on context.

Donner des frissons can describe:

• fear
• suspense
• discomfort
• strong emotion
• awe
• excitement

So the sentence might mean the story is:

• scary
• haunting
• powerful
• emotionally intense

Context determines the feeling.

Yes. A few common ones are:

• Cette histoire me fait frissonner. = This story makes me shiver.
• Cette histoire me donne la chair de poule. = This story gives me goosebumps.
• Cette histoire est glaçante. = This story is chilling.

Among these, donner des frissons is a very natural and common way to express the idea.

A rough pronunciation is:

set is-twar mə don day free-son

More closely:

/sɛt istwaʁ mə dɔn de fʁisɔ̃/

A few helpful points:

• Cette sounds like set
• histoire begins as if the h were not there
• me is usually a light muh
• des frissons links smoothly in natural speech

In careful speech, the whole sentence flows quite smoothly: Cette histoire me donne des frissons.